Concentration bounds for CVaR estimation: The cases of light-tailed and heavy-tailed distributions

Prashanth L.A., Krishna Jagannathan, Ravi Kolla
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:5577-5586, 2020.

Abstract

Conditional Value-at-Risk (CVaR) is a widely used risk metric in applications such as finance. We derive concentration bounds for CVaR estimates, considering separately the cases of sub-Gaussian, light-tailed and heavy-tailed distributions. For the sub-Gaussian and light-tailed cases, we use a classical CVaR estimator based on the empirical distribution constructed from the samples. For heavy-tailed random variables, we assume a mild ‘bounded moment’ condition, and derive a concentration bound for a truncation-based estimator. Our concentration bounds exhibit exponential decay in the sample size, and are tighter than those available in the literature for the above distribution classes. To demonstrate the applicability of our concentration results, we consider the CVaR optimization problem in a multi-armed bandit setting. Specifically, we address the best CVaR-arm identification problem under a fixed budget. Using our CVaR concentration results, we derive an upper-bound on the probability of incorrect arm identification.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-l-a-20a, title = {Concentration bounds for {CV}a{R} estimation: The cases of light-tailed and heavy-tailed distributions}, author = {L.A., Prashanth and Jagannathan, Krishna and Kolla, Ravi}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {5577--5586}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/l-a-20a/l-a-20a.pdf}, url = {https://proceedings.mlr.press/v119/l-a-20a.html}, abstract = {Conditional Value-at-Risk (CVaR) is a widely used risk metric in applications such as finance. We derive concentration bounds for CVaR estimates, considering separately the cases of sub-Gaussian, light-tailed and heavy-tailed distributions. For the sub-Gaussian and light-tailed cases, we use a classical CVaR estimator based on the empirical distribution constructed from the samples. For heavy-tailed random variables, we assume a mild ‘bounded moment’ condition, and derive a concentration bound for a truncation-based estimator. Our concentration bounds exhibit exponential decay in the sample size, and are tighter than those available in the literature for the above distribution classes. To demonstrate the applicability of our concentration results, we consider the CVaR optimization problem in a multi-armed bandit setting. Specifically, we address the best CVaR-arm identification problem under a fixed budget. Using our CVaR concentration results, we derive an upper-bound on the probability of incorrect arm identification.} }
Endnote
%0 Conference Paper %T Concentration bounds for CVaR estimation: The cases of light-tailed and heavy-tailed distributions %A Prashanth L.A. %A Krishna Jagannathan %A Ravi Kolla %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-l-a-20a %I PMLR %P 5577--5586 %U https://proceedings.mlr.press/v119/l-a-20a.html %V 119 %X Conditional Value-at-Risk (CVaR) is a widely used risk metric in applications such as finance. We derive concentration bounds for CVaR estimates, considering separately the cases of sub-Gaussian, light-tailed and heavy-tailed distributions. For the sub-Gaussian and light-tailed cases, we use a classical CVaR estimator based on the empirical distribution constructed from the samples. For heavy-tailed random variables, we assume a mild ‘bounded moment’ condition, and derive a concentration bound for a truncation-based estimator. Our concentration bounds exhibit exponential decay in the sample size, and are tighter than those available in the literature for the above distribution classes. To demonstrate the applicability of our concentration results, we consider the CVaR optimization problem in a multi-armed bandit setting. Specifically, we address the best CVaR-arm identification problem under a fixed budget. Using our CVaR concentration results, we derive an upper-bound on the probability of incorrect arm identification.
APA
L.A., P., Jagannathan, K. & Kolla, R.. (2020). Concentration bounds for CVaR estimation: The cases of light-tailed and heavy-tailed distributions. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:5577-5586 Available from https://proceedings.mlr.press/v119/l-a-20a.html.

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